“Furthermore, I find the idea that the whole is the sum of its parts quite intuitive, and infinities are not compatible with that (inf/2 = inf).”
Note that infinite sets in standard math satisfy
the whole is the sum (union) of the parts,
the size (cardinality, or with measurable sets, the measure) of the whole is the sum of the sizes of the parts,* and
the part is the whole with its complement in the whole removed.
They don’t in general satisfy the size of the difference is the difference of the sizes.
*Except for uncountable sums, where size=measure
Thanks for clarifying! I have changed the sentence you quoted to:
Furthermore, infinities are compatible with the whole being equal to its parts (e.g. inf = inf/2), and I find that quite unintuitive.
It’s not exactly the whole being equal to its parts, since they will be different sets, but they can have the same size.
Thanks, again! I have updated the sentence to:
Furthermore, infinities are compatible with the size of the whole being equal to that of each of its parts (e.g. inf = inf/2), and I find that quite unintuitive.
“Furthermore, I find the idea that the whole is the sum of its parts quite intuitive, and infinities are not compatible with that (inf/2 = inf).”
Note that infinite sets in standard math satisfy
the whole is the sum (union) of the parts,
the size (cardinality, or with measurable sets, the measure) of the whole is the sum of the sizes of the parts,* and
the part is the whole with its complement in the whole removed.
They don’t in general satisfy the size of the difference is the difference of the sizes.
*Except for uncountable sums, where size=measure
Thanks for clarifying! I have changed the sentence you quoted to:
It’s not exactly the whole being equal to its parts, since they will be different sets, but they can have the same size.
Thanks, again! I have updated the sentence to: